An Improved Approximation Algorithm for the Traveling Salesman Problem with Relaxed Triangle Inequality

TL;DR

This paper introduces an improved polynomial-time approximation algorithm for the Traveling Salesman Problem with relaxed triangle inequality, achieving better bounds by leveraging novel graph orientation techniques.

Contribution

It presents a new approximation algorithm for beta-TSP with improved bounds and a novel use of orientations in graphs, extending previous methods.

Findings

Achieves a 1.5-approximation for the standard TSP.

Develops a (3 beta/4 + 3 beta^2/4)-approximation for beta-TSP.

Introduces a polynomial-time algorithm for degree-four-bounded Eulerian subgraphs.

Abstract

Given a complete edge-weighted graph G, we present a polynomial time algorithm to compute a degree-four-bounded spanning Eulerian subgraph of 2G that has at most 1.5 times the weight of an optimal TSP solution of G. Based on this algorithm and a novel use of orientations in graphs, we obtain a (3 beta/4 + 3 beta^2/4)-approximation algorithm for TSP with beta-relaxed triangle inequality (beta-TSP), where beta >= 1. A graph G is an instance of beta-TSP, if it is a complete graph with non-negative edge weights that are restricted as follows. For each triple of vertices u,v,w in V(G), c({u,v}) <= beta (c({u,w}) + c({w,v})).